Characterizations of \(k\)-potent elements in rings. (Q2353834)

\textit{O. M. Baksalary} and \textit{G. Trenkler} have systematically investigated [in Electron. J. Linear Algebra 26, 446-470 (2013; Zbl 1283.15043)] \(k\)-potent, in particular, tripotent complex matrices. In the paper under review, the author turns to \(k\)-potent elements in rings; all rings considered are associative rings with unit and (in most cases) involution. The bulk of the paper is devoted to various characterizations of tripotent elements, mainly in terms of idempotents, EP elements, group and Moore-Penrose inverses. Some characterizations in rings with involution are given also to \(k\)-potent elements and EP elements, i.e., elements for which both the group inverse and Moore-Penrose inverse exist and are equal. Several results obtained in [op. cit.] are shown to have a direct ring-theoretical analogue.